Mixed phase space, Hamiltonian systems

The other distribution is the microcanonical equilibrium distribution. More than 15 years ago, Ott-Brown-Grebogi pointed out fractional scaling of deviation from ergodic adiabatic invariants in Hamiltonian chaotic systems [16, 17]. We will reconsider not only ergodic adiabatic invariants but also nonergodic adiabatic invariants, which are important in the mixed phase space. We will show results of our numerical simulation in which a nonergodic adiabatic invariant corresponding to uniform distribution is broken in the mixed phase space. 

In this section we will consider an nonergodic adiabatic invariant for a chaotic Hamiltonian system. Specifically, we are interested in the mixed phase space, in which tori and chaotic seas coexist. 

With regard to the microcanonical equilibrium distribution and the extension of the fluctuation-dissipation theorem, we considered a nonergodic adiabatic invariant in a simple Hamiltonian chaotic system. We numerically demonstrated the breaking of the nonergodic adiabatic invariant in the mixed phase space. The variance of the nonergodic adiabatic invariant can be considered as a measure for complexity of the mixed phase space. 

II. Anomalous Transport in Hamiltonian Systems with Mixed Phase Space... 

II. ANOMALOUS TRANSPORT IN HAMILTONIAN SYSTEMS WITH MIXED PHASE SPACE... 

A major question however is how to construct the phase diagram of mixed systems. In order to do that, one needs to enlarge the boson space, by introducing an auxiliary boson, So, that transforms as the representation Si, and represents pairs in the normal phase. The Hamiltonian becomes... 

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